Elastic jump in a quadratic pressure-area tube ($q^2/\alpha +{\alpha }^3/3=\mathrm{constant}$)

An elastic jump is a short transition from a fast, narrow supercritical collapsible tube flow to a slower, wider subcritical flow. Neglecting body force and distributed friction across the jump conserves flux and the momentum flux shown above. Distinct upstream and downstream areas therefore obey ${\alpha }_2^3+{\alpha }_1{\alpha }_2^2+{\alpha }_1^2{\alpha }_2=3q^2/{\alpha }_1$. The flux function has its unique minimum at $\alpha =\sqrt{q}$, so an upstream state below this value has one conjugate state above it. The jump dissipates mechanical energy; it does not conserve a Bernoulli constant.